# Interaction of Poisson hyperplane processes and convex bodies

## TL;DR

This paper studies how well a random polytope, formed by hyperplanes of a Poisson process that do not intersect a convex body, approximates the body's mean width as the hyperplane intensity increases.

## Contribution

It provides an analysis of the approximation quality of the random polytope to the convex body in terms of mean width, as hyperplane intensity tends to infinity.

## Key findings

- Expected mean width of the random polytope converges to that of the convex body.
- Quantitative bounds on the approximation error are derived.
- Results extend understanding of hyperplane process interactions with convex geometry.

## Abstract

Given a stationary and isotropic Poisson hyperplane process and a convex body $K$ in ${\mathbb R}^d$, we consider the random polytope defined by the intersection of all closed halfspaces containing $K$ that are bounded by hyperplanes of the process not intersecting $K$. We investigate how well the expected mean width of this random polytope approximates the mean width of $K$ if the intensity of the hyperplane process tends to infinity.